Why would the designers need to or think of googling "cogwheels in flags" in the first place?
It's just not that controversial an image.

Exactly. It's hard to make anything terribly controversial out of symbolism which stands for "cooperative effort toward a common goal" or, put more simply, "working together". If there was anything sinister about the Deutsche Arbeitsfront flag, it had much more to do with the color scheme and the swastika than with the gear wheel symbol, which was very likely adopted there for much the same reason it figured in the emblems of many of the other organizations and entities already named in this thread.

__________________"It is my belief that nearly any invented quotation, played with
confidence, stands a good chance to deceive." — Mark Twain